I did this by imagining a Venn diagram.
Number of relations which are Reflexive and Symmetric would be given by
$2^{\binom{n}{2}}$ . Now, this also contains some Anti-symmetric relations.
Number of relations which are Reflexive, Symmetric and Anti-symmetric would be $2^n$
And hence final answer must be given by $2^{\binom{n}{2}}-2^n$
Am I correct in my reasoning?
No, you are not. You should emphasize on the definition of Reflexive relations. The number of Reflexive, Symmetric, Anti-symmetric relations is only $1$ because we need to have all the diagonal elements of the grid
in our subset relations. $2^n$ $(n=6\ \text{here} )$ means that you are considering selections of the diagonal elements whether to take one or not. The answer should be $2^{15}-1$. (Possible selections of one side of non-diagonal elements with the other side corresponding non-diagonal element and all diagonal ones to be selected automatically "minus" case in which no non-diagonal elements are chosen which makes it Reflexive, Symmetric, Anti-symmetric altogether)